1. 1 Discrete Structures & Algorithms Discrete Probability

2. 2 Probability Theory: Counting in Terms of Proportions

3. 3 The Descendants of Adam Adam was X inches tall He had two sons: One was X+1 inches tall One was X-1 inches tall Each of his sons had two sons …

4. 4 X X-1X+1 X-2X+2X X-3X+3X-1X+1 X-4X+4X-2X+2X 1 11 1 1 2 1331 14641 In the n th generation there will be 2 n males, each with one of n+1 different heights: h 0, h 1,…,h n h i = (X-n+2i) occurs with proportion: nini / 2 n

5. 5 Unbiased Binomial Distribution On n+1 Elements Let S be any set {h 0, h 1, …, h n } where each element h i has an associated probability Any such distribution is called an Unbiased Binomial Distribution (or Unbiased Bernoulli Distribution) nini 2n2n

6. 6 Some Puzzles

7. 7 Teams A and B are equally good In any one game, each is equally likely to win What is most likely length of a “best of 7” series? Flip coins until either 4 heads or 4 tails Q: Is this more likely to take 6 or 7 flips?

8. 8 6 and 7 Are Equally Likely To reach either one, after 5 games, it must be 3 to 2 ½ chance it ends 4 to 2; ½ chance it doesn’t

9. 9 Silver and Gold One bag has two silver coins, another has two gold coins, and the third has one of each One bag is selected at random. One coin from it is selected at random. It turns out to be gold What is the probability that the other coin is gold?

10. 10 3 choices of bag 2 ways to order bag contents 6 equally likely paths

11. 11 Given that we see a gold, 2/3 of remaining paths have gold in them!

12. 12 So, sometimes, probabilities can be counter-intuitive ? ?

13. 13 Language of Probability The formal language of probability is a very important tool in describing and analyzing probability distribution

14. 14 Finite Probability Distribution A (finite) probability distribution D is a finite set S of elements, where each element x in S has a positive real weight, proportion, or probability p(x)  p(x) = 1 x  S For convenience we will define D(x) = p(x) S is often called the sample space and elements x in S are called samples The weights must satisfy:

15. 15 S Sample space Sample Space D(x) = p(x) = 0.2 weight or probability of x 0.2 0.13 0.06 0.11 0.17 0.1 0.13 0 0.1

16. 16 Events Any set E  S is called an event  p(x) x  E Pr D [E] = S 0.17 0.1 0.13 0 Pr D [E] = 0.4

17. 17 Uniform Distribution If each element has equal probability, the distribution is said to be uniform  p(x) = x  E Pr D [E] = |E| |S|

18. 18 A fair coin is tossed 100 times in a row What is the probability that we get exactly half heads?

19. 19 The sample space S is the set of all outcomes {H,T} 100 Each sequence in S is equally likely, and hence has probability 1/|S|=1/2 100 Using the Language

20. 20 S = all sequences of 100 tosses x = HHTTT……TH p(x) = 1/|S| Visually

21. 21 Set of all 2 100 sequences {H,T} 100 Probability of event E = proportion of E in S Event E = Set of sequences with 50 H’s and 50 T’s 100 50 / 2 100

22. 22 Suppose we roll a white die and a black die What is the probability that sum is 7 or 11?

23. 23 (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) } Pr[E] = |E|/|S| = proportion of E in S = 8/36 Same Methodology! S = {

24. 24 23 people are in a room Suppose that all possible birthdays are equally likely What is the probability that two people will have the same birthday?

25. 25 x = (17,42,363,1,…, 224,177) 23 numbers And The Same Methods Again! Sample space W = {1, 2, 3, …, 366} 23 Event E = { x  W | two numbers in x are same } Count |E| instead!What is |E|?

26. 26 all sequences in S that have no repeated numbers E = |W| = 366 23 |E| = (366)(365)…(344) = 0.494… |W| |E| |W| = 0.506…

27. 27 and is defined to be = S A B proportion of A  B More Language Of Probability The probability of event A given event B happened is written Pr[ A | B ] to B Pr [ A  B ] Pr [ B ]

28. 28 event A = {white die = 1} event B = {total = 7} Suppose we roll a white die and black die What is the probability that the white is 1 given that the total is 7?

29. 29 (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) } S = { |B| Pr[B] 1/6 |A  B| =Pr [ A | B ] Pr [ A  B ] 1/36 == event A = {white die = 1}event B = {total = 7}

30. 30 Independence! A and B are independent events if Pr[ A | B ] = Pr[ A ]  Pr[ A  B ] = Pr[ A ] Pr[ B ]  Pr[ B | A ] = Pr[ B ]

31. 31 Pr[A 1 | A 2  A 3 ] = Pr[A 1 ] Pr[A 2 | A 1  A 3 ] = Pr[A 2 ] Pr[A 3 | A 1  A 2 ] = Pr[A 3 ] Pr[A 1 | A 2 ] = Pr[A 1 ]Pr[A 1 | A 3 ] = Pr[A 1 ] Pr[A 2 | A 1 ] = Pr[A 2 ] Pr[A 2 | A 3 ] = Pr[A 2 ] Pr[A 3 | A 1 ] = Pr[A 3 ]Pr[A 3 | A 2 ] = Pr[A 3 ] E.g., {A 1, A 2, A 3 } are independent events if: Independence! A 1, A 2, …, A k are independent events if knowing if some of them occurred does not change the probability of any of the others occurring

32. 32 Silver and Gold One bag has two silver coins, another has two gold coins, and the third has one of each One bag is selected at random. One coin from it is selected at random. It turns out to be gold What is the probability that the other coin is gold?

33. 33 Let G 1 be the event that the first coin is gold Pr[G 1 ] = 1/2 Let G 2 be the event that the second coin is gold Pr[G 2 | G 1 ] = Pr[G 1 and G 2 ] / Pr[G 1 ] = (1/3) / (1/2) = 2/3 Note: G 1 and G 2 are not independent

34. 34 Monty Hall Problem Announcer hides prize behind one of 3 doors at random You select some door Announcer opens one of others with no prize You can decide to keep or switch What to do?

35. 35 Staying we win if we choose the correct door Switching we win if we choose the incorrect door Pr[ choosing correct door ] = 1/3 Pr[ choosing incorrect door ] = 2/3 Monty Hall Problem Sample space = { prize behind door 1, prize behind door 2, prize behind door 3 } Each has probability 1/3

36. 36 We are inclined to think: “After one door is opened, others are equally likely…” But his action is not independent of yours! Why Was This Tricky?

37. 37 Binomial Distribution Definition Language of Probability Sample Space Events Uniform Distribution Pr [ A | B ] Independence